analytic polyhedron


Suppose Gn is a domain and let WG be an open set. Let f1,,fk:W be holomorphic functionsMathworldPlanetmath. Then if the set


is relatively compactPlanetmathPlanetmath in W, we say that Ω is an analytic polyhedron in G. Sometimes it is denoted Ω(f1,,fk). Further (W,f1,,fk) is called the of the analytic polyhedron.

An analytic polyhedron is automatically a domain of holomorphy by using the functions that define it as g(z):=1eiθ-fj(z) to show that g cannot be extended beyond a point where fj(z)=eiθ. Every boundary point of Ω is of that form for some fj.

Furthermore every domain of holomorphy can be exhausted by analytic polyhedra (that is, every compact subset is contained in an analytic polyhedron) and in fact only domains of holomorphy can be exhausted by analytic polyhedra, see the Behnke-Stein theorem.

Note that sometimes W is required to be homeomorphic to the unit ball.


  • 1 Lars Hörmander. , North-Holland Publishing Company, New York, New York, 1973.
  • 2 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.
Title analytic polyhedron
Canonical name AnalyticPolyhedron
Date of creation 2013-03-22 14:32:39
Last modified on 2013-03-22 14:32:39
Owner jirka (4157)
Last modified by jirka (4157)
Numerical id 7
Author jirka (4157)
Entry type Definition
Classification msc 32T05
Classification msc 32A07
Synonym analytic polyhedra
Defines frame of an analytic polyhedron